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+// Copyright 2017 The Abseil Authors.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+// https://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+#include "absl/random/internal/chi_square.h"
+
+#include <cmath>
+
+#include "absl/random/internal/distribution_test_util.h"
+
+namespace absl {
+inline namespace lts_2019_08_08 {
+namespace random_internal {
+namespace {
+
+#if defined(__EMSCRIPTEN__)
+// Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
+inline double fma(double x, double y, double z) {
+ return (x * y) + z;
+}
+#endif
+
+// Use Horner's method to evaluate a polynomial.
+template <typename T, unsigned N>
+inline T EvaluatePolynomial(T x, const T (&poly)[N]) {
+#if !defined(__EMSCRIPTEN__)
+ using std::fma;
+#endif
+ T p = poly[N - 1];
+ for (unsigned i = 2; i <= N; i++) {
+ p = fma(p, x, poly[N - i]);
+ }
+ return p;
+}
+
+static constexpr int kLargeDOF = 150;
+
+// Returns the probability of a normal z-value.
+//
+// Adapted from the POZ function in:
+// Ibbetson D, Algorithm 209
+// Collected Algorithms of the CACM 1963 p. 616
+//
+double POZ(double z) {
+ static constexpr double kP1[] = {
+ 0.797884560593, -0.531923007300, 0.319152932694,
+ -0.151968751364, 0.059054035642, -0.019198292004,
+ 0.005198775019, -0.001075204047, 0.000124818987,
+ };
+ static constexpr double kP2[] = {
+ 0.999936657524, 0.000535310849, -0.002141268741, 0.005353579108,
+ -0.009279453341, 0.011630447319, -0.010557625006, 0.006549791214,
+ -0.002034254874, -0.000794620820, 0.001390604284, -0.000676904986,
+ -0.000019538132, 0.000152529290, -0.000045255659,
+ };
+
+ const double kZMax = 6.0; // Maximum meaningful z-value.
+ if (z == 0.0) {
+ return 0.5;
+ }
+ double x;
+ double y = 0.5 * std::fabs(z);
+ if (y >= (kZMax * 0.5)) {
+ x = 1.0;
+ } else if (y < 1.0) {
+ double w = y * y;
+ x = EvaluatePolynomial(w, kP1) * y * 2.0;
+ } else {
+ y -= 2.0;
+ x = EvaluatePolynomial(y, kP2);
+ }
+ return z > 0.0 ? ((x + 1.0) * 0.5) : ((1.0 - x) * 0.5);
+}
+
+// Approximates the survival function of the normal distribution.
+//
+// Algorithm 26.2.18, from:
+// [Abramowitz and Stegun, Handbook of Mathematical Functions,p.932]
+// http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
+//
+double normal_survival(double z) {
+ // Maybe replace with the alternate formulation.
+ // 0.5 * erfc((x - mean)/(sqrt(2) * sigma))
+ static constexpr double kR[] = {
+ 1.0, 0.196854, 0.115194, 0.000344, 0.019527,
+ };
+ double r = EvaluatePolynomial(z, kR);
+ r *= r;
+ return 0.5 / (r * r);
+}
+
+} // namespace
+
+// Calculates the critical chi-square value given degrees-of-freedom and a
+// p-value, usually using bisection. Also known by the name CRITCHI.
+double ChiSquareValue(int dof, double p) {
+ static constexpr double kChiEpsilon =
+ 0.000001; // Accuracy of the approximation.
+ static constexpr double kChiMax =
+ 99999.0; // Maximum chi-squared value.
+
+ const double p_value = 1.0 - p;
+ if (dof < 1 || p_value > 1.0) {
+ return 0.0;
+ }
+
+ if (dof > kLargeDOF) {
+ // For large degrees of freedom, use the normal approximation by
+ // Wilson, E. B. and Hilferty, M. M. (1931)
+ // chi^2 - mean
+ // Z = --------------
+ // stddev
+ const double z = InverseNormalSurvival(p_value);
+ const double mean = 1 - 2.0 / (9 * dof);
+ const double variance = 2.0 / (9 * dof);
+ // Cannot use this method if the variance is 0.
+ if (variance != 0) {
+ return std::pow(z * std::sqrt(variance) + mean, 3.0) * dof;
+ }
+ }
+
+ if (p_value <= 0.0) return kChiMax;
+
+ // Otherwise search for the p value by bisection
+ double min_chisq = 0.0;
+ double max_chisq = kChiMax;
+ double current = dof / std::sqrt(p_value);
+ while ((max_chisq - min_chisq) > kChiEpsilon) {
+ if (ChiSquarePValue(current, dof) < p_value) {
+ max_chisq = current;
+ } else {
+ min_chisq = current;
+ }
+ current = (max_chisq + min_chisq) * 0.5;
+ }
+ return current;
+}
+
+// Calculates the p-value (probability) of a given chi-square value
+// and degrees of freedom.
+//
+// Adapted from the POCHISQ function from:
+// Hill, I. D. and Pike, M. C. Algorithm 299
+// Collected Algorithms of the CACM 1963 p. 243
+//
+double ChiSquarePValue(double chi_square, int dof) {
+ static constexpr double kLogSqrtPi =
+ 0.5723649429247000870717135; // Log[Sqrt[Pi]]
+ static constexpr double kInverseSqrtPi =
+ 0.5641895835477562869480795; // 1/(Sqrt[Pi])
+
+ // For large degrees of freedom, use the normal approximation by
+ // Wilson, E. B. and Hilferty, M. M. (1931)
+ // Via Wikipedia:
+ // By the Central Limit Theorem, because the chi-square distribution is the
+ // sum of k independent random variables with finite mean and variance, it
+ // converges to a normal distribution for large k.
+ if (dof > kLargeDOF) {
+ // Re-scale everything.
+ const double chi_square_scaled = std::pow(chi_square / dof, 1.0 / 3);
+ const double mean = 1 - 2.0 / (9 * dof);
+ const double variance = 2.0 / (9 * dof);
+ // If variance is 0, this method cannot be used.
+ if (variance != 0) {
+ const double z = (chi_square_scaled - mean) / std::sqrt(variance);
+ if (z > 0) {
+ return normal_survival(z);
+ } else if (z < 0) {
+ return 1.0 - normal_survival(-z);
+ } else {
+ return 0.5;
+ }
+ }
+ }
+
+ // The chi square function is >= 0 for any degrees of freedom.
+ // In other words, probability that the chi square function >= 0 is 1.
+ if (chi_square <= 0.0) return 1.0;
+
+ // If the degrees of freedom is zero, the chi square function is always 0 by
+ // definition. In other words, the probability that the chi square function
+ // is > 0 is zero (chi square values <= 0 have been filtered above).
+ if (dof < 1) return 0;
+
+ auto capped_exp = [](double x) { return x < -20 ? 0.0 : std::exp(x); };
+ static constexpr double kBigX = 20;
+
+ double a = 0.5 * chi_square;
+ const bool even = !(dof & 1); // True if dof is an even number.
+ const double y = capped_exp(-a);
+ double s = even ? y : (2.0 * POZ(-std::sqrt(chi_square)));
+
+ if (dof <= 2) {
+ return s;
+ }
+
+ chi_square = 0.5 * (dof - 1.0);
+ double z = (even ? 1.0 : 0.5);
+ if (a > kBigX) {
+ double e = (even ? 0.0 : kLogSqrtPi);
+ double c = std::log(a);
+ while (z <= chi_square) {
+ e = std::log(z) + e;
+ s += capped_exp(c * z - a - e);
+ z += 1.0;
+ }
+ return s;
+ }
+
+ double e = (even ? 1.0 : (kInverseSqrtPi / std::sqrt(a)));
+ double c = 0.0;
+ while (z <= chi_square) {
+ e = e * (a / z);
+ c = c + e;
+ z += 1.0;
+ }
+ return c * y + s;
+}
+
+} // namespace random_internal
+} // inline namespace lts_2019_08_08
+} // namespace absl
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